Green’s function

green

The Green’s function is a concept developed by George Green in the XIXth century. It is useful to solve inhomogenous differential equations such as :

\displaystyle \frac{dy}{dt} = f(t)y(t)+g(t)

where the functions f and g are defined on an interval.

GREEN’S FUNCTION

More generally, consider an equation with a linear differential operator L :

\displaystyle Ly = g

Green’s function

A Green’s function G(t,s) of a linear operator L is any solution of the equation

\displaystyle LG(t,s) = \delta(t-s)

where \delta(t-s) is the Dirac delta-function.

The original equation can be rewritten as follows :

\displaystyle Ly(t) = g(t)

\displaystyle Ly(t) = \int \delta(t-s) g(s) ds

\displaystyle Ly(t) = \int LG(t,s) g(s) ds

\displaystyle Ly(t) = L \int G(t,s) g(s) ds

\displaystyle y(t) = \int G(t,s) g(s) ds

In this way, the Green’s function plays the role of an integral kernel in the solution of the equation Ly=g.

INHOMOGENEOUS LINEAR EQUATIONS

A first order inhomogenous linear equation is an equation

\displaystyle \frac{dy}{dt} = f(t)y(t)+g(t)

It can be solved with initial condition y(t_0) = 0 :

\displaystyle \frac{d}{dt}\left( y(t) e^{\int_{t_0}^{t} -f} \right) = \frac{dy}{dt} e^{\int_{t_0}^{t} -f} - y(t) f(t) e^{\int_{t_0}^{t} -f}

\displaystyle \frac{d}{dt}\left( y(t) e^{\int_{t_0}^{t} -f} \right) = g(t) e^{\int_{t_0}^{t} -f}

\displaystyle y(t) e^{\int_{t_0}^{t} -f} = \int_{t_0}^{t} g(\xi) e^{\int_{t_0}^{\xi} -f} d\xi

\displaystyle y(t) = \int_{t_0}^{t} g(\xi) e^{\int_{t_0}^{t} f - \int_{t_0}^{\xi} f} d\xi

\displaystyle y(t) = \int_{t_0}^{t} g(\xi) e^{\int_{\xi}^{t} f} d\xi

GREEN’S FUNCTION OF A INHOMOGENOUS LINEAR EQUATION

What is the Green’s function of a first order inhomogenous linear equation ?

The Green’s function satisfies the equation

\displaystyle \frac{d}{dt}G(t,s) - f(t)G(t,s) = \delta(t-s)

Then, we apply the solution defined above of a inhomogenous linear equation with the condition t_0<s<t

\displaystyle G(t,s) = \int_{t_0}^{t} \delta(\xi-s) e^{\int_{\xi}^{t} f} d\xi

\displaystyle G(t,s) = e^{\int_{s}^{t} f}

EXAMPLE

We illustrate the Green’s function on the following equation

\displaystyle y' = y+1

The Green’s function is computed as

\displaystyle G(t,s) = e^{\int_{s}^{t} f} = e^{t-s}

It is graphically represented at the beginning of the post.

We deduce the solution

\displaystyle y(t) = \int_{t_0}^{t} e^{t-s}g(s)ds

\displaystyle y(t) = \int_{t_0}^{t} e^{t-s}ds

\displaystyle y(t) = e^{t-t_0}-1

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